ADC 2014 Applied Demography Conference San Antonio 1/8-10/14 Understanding Spatial Spillovers & Feedbacks Paul R. Voss University of North Carolina at Chapel Hill and André Braz Golgher University of Minas Gerais (UFMG), Brazi l

Spatial Effects ADC 2014 Spatial spillovers Spatial feedbacks

ADC 2014 Spatial Lag Model (SAR model) Model assumes large-scale spatial heterogeneity is handled by Xβ ; remaining small-scale (localized) spatial dependence is handled as an autoregressive, interactive, effect through Wy & 

ADC 2014 Rearranging the terms in this model we obtain the reduced form: Not an easy model to explain in words

ADC 2014 Fortunately… So we can re-express our model… and taking expected values…

ADC 2014 Example: Counties in U.S. South, 2000 Census Source: SF3 Table P87 n = 1,387

Standard Linear Model (transformed variables) ADC 2014 For this model, we all know how to interpret the fixed marginal effects

But now… Spatial Lag Model (SAR) (transformed variables) ADC 2014 For this model, interpreting marginal effects is much more difficult avg. of “neighbor’s” child poverty rate I’ll return to this model at the end of my presentation

ADC 2014 Why more difficult? recall expected values for SAR model… Spatial Spillovers & Spatial Feedbacks and what does this mean?

ADC 2014 Illustration of “spatial spillover” on child poverty from a simulated increase in female-headed HH (with kids) only in Autauga Co. AL (SAR model with 1 st -order queen W ) Effect of arbitrary increase in FEMHH rate in Autauga Co. on Child Poverty Rate (SAR Model)

ADC 2014 The next several slides attempt to show (mathematically) what’s happening here

ADC 2014 Back to the Standard Linear Model (SLM) In matrix notation: Q: For the standard linear model, what (mathematically) do the beta coefficients represent? A: They represent the partial derivatives of y with respect to x k. and a reminder:  ~ N iid (0, σ 2 I )

ADC 2014 Think of it like this… In matrix notation: (n x n)(n x n) (n x n)(n x n) (n x n)(n x n) one of these for each β coefficient; i.e., one for each x k

ADC 2014 That is… cross-partial derivatives are zero; change in x ik affects only y i ; no spatial spillovers row 1: change in DV in region 1 based on change in x ik each region ( i = 1,…, n )

ADC 2014 and… cross-partial derivatives are zero; change in x ik affects only y i ; no spatial spillovers column 1: change in DV in each region i (i = 1,…,n) b ased on change in x 1k (i.e., x k in region 1)

ADC 2014 Simple example… Region 1 Region 2 Region 3 Assume: 3 regions; independent observations Standard linear model (SLM) estimated by OLS Dependent variable y ; Independent variable x 1 Slope coefficient for x 1, β 1 = 2 Partial derivatives matrix for x 1 :

ADC 2014 Simple example (continued) … Region 1 Region 2 Region 3 Now, further assume that variable x 1 in Region 3 changes by 10 units: Only the value of y 3 is affected by the change in x 3 ; no spatial spillovers

ADC 2014 Again… things are more complex for SAR model (spatial spillovers!)

ADC 2014 Now the partial derivative matrix looks like this… The n x n matrix ( I-  W ) -1 has non-zero elements off the main diagonal. These non-zero cross-partial derivatives imply the existence of spatial spillovers. This follows from the power series approximation shown above:

ADC 2014 With W row-standardized, the elements of W lie between 0 & 1. Further, under positive spatial autocorrelation,  is constrained to be strictly < |1|. Thus, the spatial spillovers associated with higher powers of  & W dampen out, often quickly. Again, assume: Region 1 Region 2 Region 3 Further, assume the row-standardized W matrix:

ADC 2014 and its inverse: For this particular weights matrix, we have (by simple matrix arithmetic):

ADC 2014 Thus: The n x n matrix of partial derivatives is a function of the exogenously specified weights matrix, W, the spatial scalar parameter, , and the parameter β k. But the β -term appears not only along the major diagonal but also in the cross- partial derivatives. When  = 0, the matrix of partial derivatives is the one we saw for the SLM. When   0, the partial derivatives (diagonal) are greater than the β -term, augmented by spatial spillovers as follows:

ADC 2014 Diagonal terms of the partial derivatives matrix (for our 3-region example and the specified 1 st -order weights matrix, W ): Again, when  = 0, the diagonal partial derivatives are simply the β -terms. When   0, the diagonal elements represent the β -terms, augmented by spatial spillovers

ADC 2014 Simple example revisited… Region 1 Region 2 Region 3 Assume: 3 regions; independent observations SAR model Dependent variable y ; Independent variable x 1 Slope coefficient for x 1, β 1 = 2 Partial derivatives matrix for x 1 :

ADC 2014 Simple example (continued) … Region 1 Region 2 Region 3 Further, assume an increase by 10 units the value of x 1 in Region 3:

ADC 2014 When  = 0, we obtain the earlier results for the SLM (i.e., increase of 10 units for variable x 1 in Region 3 results in y 3 increasing by 20 units):

ADC 2014 If, however,  = 0.2, then we see that an increase of 10 units for variable x 1 in Region 3 results in y increasing in all regions:

ADC 2014 Can we describe this outcome? y increases everywhere because of spatial spillovers. The spillover effect is strongest for Region 2 (an immediate neighbor of Region 3 (which we might have anticipated because of the 1 st -order spatial weights matrix, W ). But Region 1 is a “neighbor of the neighbor” and also is affected by the change in Region 3 for this SAR model. But why did y in Region 3 increase by more than 20? The answer is feedback spillover. A portion of the change in the other regions is feeding back to further change y 3.

ADC 2014 Calculate the extent of spatial spillover for this simple SAR model by subtracting the results for  = 0.0 from the results for  = 0.2 The change in Region 3 spills over to affect Region 2 and, much more mildly, even Region 1. Since Region 3 is also a neighbor or Region 2, the same degree of spillover comes back to Region 3 as feedback spillover

For the southeastern counties, and for the simple model & estimated parameters shown earlier (3 independent variables plus spatial lag)… ADC 2014 Employing the SAR model, we chose arbitrarily to increase the independent variable FEMHH in Autauga County AL by 20 percentage points (from 19% to 39%) and inquired about the impact of this change on county-level poverty rates

Change in child poverty rate due to simulated increase in FEMHH variable in Autauga Co. ADC 2014

Spatial spillovers due to change in child poverty rate due to simulated increase in FEMHH variable in Autauga Co. (direct effect removed)

ADC 2014 Simulated spatial effects of change in child poverty rate due to increase in femhht variable in Autauga Co.

Graph of spatial spillovers ADC st -order neighbors 2 nd -order neighbors 3 rd -order neighbors

Comments… Interpreting marginal effects in spatial regression models becomes complicated They are influenced by effects direct & indirect (spatial spillovers & feedbacks) They are a function of: –Our data and estimated parameters ( β k & ρ ) –Our assumed neighborhood definition and spatial weights matrix, W An example such as shown here is helpful for understanding what’s going on in these models but requires some cautions ADC 2014